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arXiv:cond-mat/0512369 · Communications in Theoretical Physics

A Direct Calculation of Critical Exponents of Two-Dimensional Anisotropic Ising Model

Xiong Gang, X. R. Wang

Published May 1, 2006• 1 citation
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Abstract

Using an exact solution of the one-dimensional quantum transverse-field Ising model, we calculate the critical exponents of the two-dimensional anisotropic classical Ising model (IM). We verify that the exponents are the same as those of isotropic classical IM. Our approach provides an alternative means of obtaining and verifying these well-known results.

DOI
10.1088/0253-6102/45/5/033

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Using an exact solution of the one-dimensional quantum transverse-field Ising model, we calculate the critical exponents of the two-dimensional anisotropic classical Ising model (IM). We verify that the exponents are the same as those of isotropic classical IM. Our approach provides an alternative means of obtaining and verifying these well-known results.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin S=12,1,32,and2S=\frac{1}{2}, 1, \frac{3}{2}, \mathrm{and} 2S=21​,1,23​,and2. The critical fields and "thermal" and correlation exponents are calculated as a function of SSS by the finite-size scaling method. The universality of the transition with respect to SSS is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.
The quantum-classical crossover critical behaviour of the Ising model in a transverse field Gamma is first studied in one dimension where exact results are obtained. The correspondence between the quantum-classical crossover behaviour of this d-dimensional quantum model and the finite-size crossover behaviour of an equivalent (d+1)-dimensional classical model exact for d=1 is extended to higher dimensions. The crossover depends strongly on the relative magnitude of the crossover exponent phi and of the critical line-shift exponent Psi . In particular, if Pi > Psi approaches the critical line along a path of constant transverse field less than the critical field Gamma c and a low temperature, only classical behaviour is seen, quantum effects do not show up. There is some controversy about the value of Psi . Experiments are suggested to give an answer.
The anisotropic XY model has a basic difference from the isotropic XY model but it behaves very similarly to the Ising model. The scaling field also changes when anisotropy is introduced. If the scaling field of an Ising model is used for the anisotropic XY model, the critical exponents are found to vary with anisotropy. Here we have shown that a proper choice of the scaling field leads to the universality and the critical exponents have the same values as in the Ising model. The scaling field is further modified to include the isotropic limit. \textcopyright{} 1996 The American Physical Society.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.